SOFTWARE FOR MULTILEVEL ANALYSIS 1
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1. SOFTWARE FOR MULTILEVEL ANALYSIS 15 with many examples Dixon and Merdian 1992 with a 5 25 floppy in the back 4 1 3 Model BMDP5 V is described in the documentation as a program for unbalanced repeated measures models with structured covariance matri ces The means that the emphasis is not on general hierarchies but specifi cally on repeated measures i e on repetitions nested within subjects To maintain the same notation as we have used before we rewrite the model as 8 Yi Vig Tijpbp Eiz where index j now stands for subjects and index 7 stands for the repeated measures nested within subjects Thus it seems there are no random coef ficients BMDP5 V allows a large number of possible choices for the within subject covariance matrices 1 which have elements 9 Ojik E j x We briefly describe the available options in matrix notation 10a D o 10b Xj w eed 07 10c X ZZ oI 10d bj AA Y 10e D OG 0 G 10f Dp Moreover there are two options that can best be defined in elementwise notation 10g He o pe 10h Ojik r Where r i k 1 Clearly this covers many of our well known friends from earlier sections Model 10a defines independent observations within subjects there is always between subject independence while model 10b sometimes de scribed as compound symmetry is in fact the random intercept model Model 10c is the2. tical packages for hierarchical linear regression GENMOD HLM ML2 VARCL Preprint 50 UCLA Statistics Los Angeles CA 1990 LG G Kreft J de Leeuw and R van der Leeden Review of five multi level analysis programs BMDP 5V GENMOD HLM ML3 VARCL American Statistician 48 324 335 1994 R C Littell G A Milliken W W Stroup and R D Wolfinger SAS System for Mixed Models SAS Institute 1996 N T Longford VARCL software for variance component analysis of data with nested random effects maximum likelihood Technical report Ed ucational Testing Service Princeton NJ 1990 N T Longford A quasi likelihood adaptation for variance component anal ysis In Poceedings of the Section on Statistical Computing of the ASA 1988 N T Longford Random Coefficient Models Oxford University Press Ox ford GB 1993 J Rasbash and Geoff Woodhouse MLn Command Reference Technical report Institute of Education University of London 1995 SAS SAS STAT software Changes and enhancements Technical report SAS Institute 1992 M Schluchter BMDP 5 V Unbalanced Repeated Measures Models with Structured Covariance Matrices Technical Report 86 BMDP Statistical Software Los Angeles CA 1988 J Singer Notes on using PROC MIXED Journal of Educational and Be havioral Statistics page 1998 BMDP Statistical Software BMDP User s Digest University of California Press 1992 22 JAN DE LEEUW
3. as a research tool to analyze the examples in a research paper or to analyze the data sets in a research project Other software is written as a commer cial product or at least it is clearly intended for general use Many programs start out in the research phase and are subsequently pro moted to the production phase Many programs are still somewhere in between Stand alone versus Module There are multilevel programs that stand on their own i e they are executables and they do not require other software except the operating system of course to be present Other programs are modules of an existing package Usually they require the rest of the package to be present as well Program versus Macro For programs written on top of an existing package there are still two possibilities Either the program is a mod ule existing in object code within the package or the program is a macro written in the scripting or extension language of the package Macros are handled by an interpreter that is part of the package SOFTWARE FOR MULTILEVEL ANALYSIS 3 Another distinction we make is between minor and major specialized pro grams This is somewhat subjective but the general idea is that a research program usually with a rather primitive interface is a minor specialized program As soon as the authors start thinking about user friendliness mak ing a nice graphical user interface adding options that people have asked for and going commercia
4. by clicking and pasting Obviously there is no way to do this in DOS so this defines is a clear distinction between MLn and MLwiN On the other hand it is unclear if this is actually superior to written output that the user herself has to read translate into a formula It seems that if users know what they are doing they can make the translation very easily themselves If they don t know what they are doing then they should not be using the program in the first place 2 1 5 Algorithm MLwiN uses the IGLS or the RIGLS algorithms first de scribed respectively by Goldstein 1986 and Goldstein 1989 The algorithms are block relation algorithms There are two blocks of parameters the fixed regression coefficients and the variance covariance 2 As in our book we use the convention that random variables are underlined 6 JAN DE LEEUW AND ITA G G KREFT components The algorithm fixes the variance components at some initial value and maximizes the likelihood over the fixed coefficients This is just a Generalized Least Squares problem Then it fixes the coefficients at their current values and maximizes the likelihood over the variance components by solving another more complicated Generalized Least Squares problem The two optimizations are alternated until convergence A concise descrip tion appears in Goldstein and Rasbash 1992 It is not entirely clear from the documentation what happens in boundary cases when dispersi
5. group statistics to compute separate regressions for each group if the groups are big enough Both VARCL and HLM use within group statistics to compute initial estimates of the variance and covariance components In our experience documented in our previous review papers the scoring algorithms is both fast and reliable 3 1 6 Extensions VARCL was also the first multilevel program that could deal with hierarchical generalized linear models Early on in the interactive questions and answers session the user can choose the error model to be either normal binomial Poisson or gamma Again quasi likelihood meth ods are used to fit these models using the procedures designed by Longford 1988 for this purpose 3 2 MIXFOO The MIXFOO program is special because it does not exist It is the generic name for a whole series of multilevel programs At the MIXFOO website we find The statistical research presented in this homepage is based on the collaborative effort of Donald Hedeker and Robert D 4in terms of the number of variables SOFTWARE FOR MULTILEVEL ANALYSIS 11 Gibbons of the University of Illinois at Chicago The com puter programs were written by Don Hedeker with interfaces written by Dave Patterson Discerning Systems Inc The names of all programs in the series start with MIX There are DOS pro grams MIXOR MIXREG MIXGSUR MIXNO and MIXPREG while MIXOR and MIXREG also exist with Windows interfaces The we
6. it is in a very remote section of their catalog indy 1 gamma rug nl sibweb catalog Catalo10 htm I12 The price is 350 250 educational 3 1 2 Documentation The distribution comes with a 100 page user man ual Longford 1990 with an additional 100 pages of example runs The manual follows the interactive interface closely but 20 pages are used for explaining some of the technical background 3 1 3 Model There are two different versions of VARCL available The first one is VARCL3 which handles three level models the second is VARCL9 which handles random intercept models with up to nine levels The model in VARCL3 is 4a Yijn By pois oh By pt ris Torit Bo Lpigh Ligh where 4b b Z sjh Bs Usp T Usjh Again we use the notation of MLwiN and we assume that oijn 1 for all i j h In VARCL9 the model is for four levels as an example 5a Yijng T Ging T PiTiijhg BpTpijhg Cijghs where 5b Oih AEU Eng E Uns Here we have singled out the intercept more explicitly by using a for the corresponding regression coefficient We see that VARCL does not have the emphasis on cross level interac tions or on two level specification that HLM has It also does not have the possibility of random coefficients on the first level which we have in MLwiN 10 JAN DE LEEUW AND ITA G G KREFT 3 1 4 Interface VARCL has acommand line interface The programs asks a large number of questions and it uses th
7. not make sense for them to learn a new language or to buy a new software package If you are already driving around in a truck you might as well pick up your groceries 5 CONCLUSIONS We have already commented throughout the text on some of the differ ences There are multilevel random coefficient and mixed linear model programs There are package modules and stand alone programs There are research and production programs One obvious characteristic they have in common is that they are almost all DOS trying to become Windows We do not really want to get too deeply into the discussion of the relative advantages of putting a GUI on top of a DOS program Clearly there are some definite advantages Graphics output becomes much more attractive and more flexible and the equations window of MLn is a useful addition which would be impossible under DOS There are also some disadvantages SOFTWARE FOR MULTILEVEL ANALYSIS 19 which are the same as for other programs with the same history such as SAS SPSS and BMDP We are forced to leave the safe domain of objective and impartial software comparison here in order to voice an opinion The GUI of all these programs is ugly as sin This is partly due to the inherent ugliness of the Windows environment but also to the fact that the interface is an afterthought Its only job is to prepare parameter and data files that are then passed to the DOS programs runnin
8. the Royal Statistical Society A 149 1 43 1986 A S Bryk and S W Raudenbush Hierachical Linear Models for So cial and Behavioral Research Applications and Data Analysis Methods Sage Publications Newbury Park CA 1991 A S Bryk S W Raudenbush and Jr R T Congdon HLM Hierarchi cal Linear and Nonlinear Modeling with the HLM 2L and HLM 3L Pro grams 1996 F M T A Busing E Meier and R van der Leeden MLA software for mul tilevel analysis of data with two levels Technical report Department of Psyhcometrics and Research Methodology University of Leiden 1994 J de Leeuw and I G G Kreft Questioning multilevel models Journal of Educational and Behavioral Statistics 20 171 190 1995 W J Dixon and K Merdian ANOVA and Regression with BMDP5 V Dixon Statistical Associates Los Angeles 1992 W J Dixon BMDP Statistical Software Manual University of California Press 1992 H Goldstein Multilevel mixed linear model analysis using iterative gener alized least squares Biometrika 73 43 56 1986 H Goldstein Multilevel Models in Educational and Social Research Grif fin London GB 1987 H Goldstein Restricted unbiased iterative generalized least squares esti mation Biometrika 76 622 623 1989 H Goldstein Multilevel Statistical Models Edward Arnold London GB 1995 H Goldstein and J Rasbash Efficient computational procedures for the estimation of parameters in multilevel models bas
9. AND ITA G G KREFT R van der Leeden K Vrijburg and J de Leeuw A review of two different approaches for the analysis of growth data using longitudinal mixed lin ear models Comparing hierarchical linear regression ML3 HLM and repeated measures design with structured covariance matrices BMDP 5V Preprint 98 UCLA Statistics Los Angeles 1991 R van der Leeden K Vrijburg and J de Leeuw A review of two differ ent approaches for the analysis of growth data using longitudinal mixed linear models Computational Statistics and Data Analysis 21 583 605 1996 G Woodhouse A guide to MLn for new users Technical report Multilevel Models Project Instutute of Education University of London 1995 M Yang J Rasbash and H Goldstein MLwiN macros for advanced mul tilevel modelling Technical report Institute of Education University of London 1998 JAN DE LEEUW UNIVERSITY OF CALIFORNIA AT LOS ANGELES E mail address deleeuw stat ucla edu ITA G G KREFT CALIFORNIA STATE UNIVERSITY LOS ANGELES E mail address kreft stat ucla edu
10. D allows for uncorrelated compound symmetry unstructured autoregressive and spatial or a block diagonal version of these which is how the levels come in again There is no factor analytic or general linear covariance structure It is interesting that HLM and MLA are true multilevel models in the sense that they specify their regression model at multiple levels MIXFOO and MLwiN have multiple levels but only a single regression equation The same is true for BMDP 5 V where the levels are defined by repeated mea sures In PROC MIXED the levels have disappeared and they have to be introduced by suitably arranging the input an parameter files This is pre cisely the reason why a paper such as Singer 1998 is necessary for some groups of users 4 2 4 Interface The interface to PROC MIXED is the familiar SAS inter face or more precisely any one of the familiar SAS interfaces As in the case of BMDP all these interfaces are rather desperate attempts to get away 18 JAN DE LEEUW AND ITA G G KREFT from the mainframe or DOS heritage Most of the windowing is used to con struct batch files with options and instructions which are then submitted to the old familair SAS engine in the background 4 2 5 Algorithm PROC MIXED can use both REML and FIML estima tion and it maximizes the likelihood by a combination of Fisher scoring and Newton Raphson The default is to use only Newton Raphson bu
11. SOFTWARE FOR MULTILEVEL ANALYSIS JAN DE LEEUW AND ITA G G KREFT CONTENTS 1 Introduction 1 1 1 Programs Macros and Packages 2 1 2 Omissions 3 2 Major Specialized Programs 3 2 1 MLwiN 4 2 2 Extensions 6 2 3 HLM 7 3 Minor Specialized Program 9 3 1 VARCL 9 3 2 MIXFOO 10 3 3 MLA 13 4 Modules in a Major Package 14 4 1 BMDP5 V 14 4 2 PROC MIXED 16 5 Conclusions 18 References 20 1 INTRODUCTION In this paper we review some of the more important software programs and packages that can are designed for or can be used for multilevel anal ysis These programs differ in many respects Some are part of major Statistics packages such as SAS or BMDP Others are written in the macro language of a major package And some are stand alone special purpose programs that can do nothing but multilevel analysis We have been involved in a number of these comparisons before The first one Kreft et al 1990 comparing HLM ML3 VARCL BMDP5 V and GENMO D was published in Kreft et al 1994 The second compari son van der Leeden et al 1991 comparing HLM ML3 and BMDP 5 V on repeated me give both th asures data was published in van der Leeden et al 1996 We e reference to the internal report version and to the published 1 2 JAN DE LEEUW AND ITA G G KREFT version because the unpublished version is usually has much more material Giving both references also shows the unfor
12. an excellent reputation because its approximately 40 programs were developed in close cooperation with excellent statisticians But BMDP has a stormy recent past The company was bought by SPSS in 1996 It is difficult to find BMDP on the main SPSS web site at http www spss com After some search it turns out that many of the BMDP products have been declared dead by SPSS and the only product still sold seems to be BMDP Classic for DOS See www spss com software science BMDP Some additional research suggests that SPSS plans to integrate BMDP in its SYSTAT product In corporate speak I would like to welcome BMDP customers to the SYSTAT family In Europe the situation is different One can buy BMDP from Statistical Solutions in Cork and their web page clearly indicates they consider it to be one of their major products 4 1 1 Availability BMDP can be ordered from www statsol ie bmdp html The price is 895 695 Academic but of course this is for the whole package all 40 routines of which BMDP 5 V is only a single one 4 1 2 Documentation The theory behind BMDP5 V is explained in Jen nrich and Schluchter 1986 Very useful information about both the method ology and the program is in Schluchter 1988 There is also a chapter on 5V in the BMDP user s manual Dixon 1992 and the BMDP user s digest Soft ware 1992 And last but not least there is a detailed how to use book
13. and not many examples but the instruc tions and options are explained in detail There is also an excellent and extensive book by Littell et al 1996 It has chapters on randomized blocks repeated measures multilevel mod els analysis of covariance spatial models heteroscedastic models and best linear unbiased prediction There are also chapters on generalized linear mixed models and nonlinear mixed models where the SAS IML macros GLIMMIX and NLINMIX are explained The book contains many exam ples worked out in detail For people who have started out with a program such as HLM which has a much simpler interface and a much smaller menu of models the use of PROC MIXED is explained expertly by Singer 1998 In terms of documentation PROC MIXED is the clear winner in our com parison In most other repects it is not 4 2 3 Model The model in PROC MIXED is very similar to the model in BMDP 5 V It may even be true historically that BMDP 5 V inspired PROC MIXED The main difference however is that the PROC MIXED specifica tion does not have levels or repeated measures They start from the general mixed model which is 11 Y Pita BpCir ty Zi Ugis E The random effects u have a covariance matrix 2 and the errors e have a covariance A As in BMDP 5 V we can specify additional parametric struc ture in these covariance matrices PROC MIXE
14. bsite also has a SAS IML macro that fit a random intercept version of MIXREG and SPSS MATRIX macros for random intercept versions of both MIXOR and MIXREG All this is freely available from the website We indicate briefly what these programs do MIXREG fits the linear mul tilevel model but it allows for various forms of autocorrelation between the first level disturbances MIXOR adds ordinal multicategory outcomes MIXGSUR does grouped time survival data MIXNO does nominal multicat egory data and MIXPREG does multilevel Poisson regression 3 2 1 Availability Software and manuals can be downloaded from www uic edu hedeker mix html All programs macros and manuals are free There are PowerMac and Sun Solaris version of the software at www stat ucla edu deleeuw mixfoo It must be emphasized that this is quite unique all other programs except VARCL and MLA only exist for DOS or Windows 3 2 2 Documentation The two core programs MIXOR and MIXREG are described in Hedeker and Gibbons 1996a and Hedeker and Gibbons 1996b These are published versions of the manuals the manuals themselves are in cluded in the software distribution The theory is described in great detail in Hedeker 1989 and for MIXOR in the article by Hedeker and Gibbons 1994 MIXGSUR MIXNO and MIXPREG also have a manuals on the website The theory of MIXGSUR is described in the technical
15. e information provided by the user to build up a setup file that describes the analysis It first builds up infor mation about the maximal model which is the largest model that can be fitted in a session Then additional questions are used to exclude variables from the maximal model to define the model to be fitted Constraints on the parameters can be defined by setting them to fixed values The model is shown and if the user likes it the program computes the estimates Then the interface asks if additional models will be fitted and if so which variables are to be removed and added and which parameters have to be constrained or freed If you know from experience which questions the program will ask and which answers you are going to give then you can put the answers in a batch file and take standard input from that file 3 1 5 Algorithm The program uses the scoring method on the reduced form It is based on the fact that within group means variances and covari ances are jointly sufficient for the model parameters Thus if we compute the within group statistics for the maximal model we can forget about the original data Since these within group statistics can be computed in the input loop the number of observations that VARCL can handle is infinitely large as long as they come in a finite number of groups This is differ ent from MLwiN which keeps the data in core for the whole session It is similar to HLM which uses the within
16. ed on iterative gener alized least squares Computational Statistics and Data Analysis 13 63 71 1992 H Goldstein J Rasbash I Plewis D Draper W Browne M Yang G Woodhouse and M Healy A user s guide to MLwiN Technical report Institute of Education University of London 1998 D Hedeker Random regression models with autocorrelated errors PhD thesis University of Chicago 1989 SOFTWARE FOR MULTILEVEL ANALYSIS 21 D Hedeker and R D Gibbons A random effects ordinal regression model for multilevel analysis Biometrics 50 933 944 1994 D Hedeker and R D Gibbons MIXOR a computer program for mixed effects ordinal probit and logistic regression analysis Computer Methods and Programs in Biomedicine 49 157 176 1996a D Hedeker and R D Gibbons MIXREG a computer program for mixed effects regression analysis with autocorrelated errors Computer Methods and Programs in Biomedicine 49 229 252 1996b D Hedeker O Siddiqui and F B Hu Random effects regression analysis of correlated grouped time survival data Technical report University of Illinois Chicago 1996 R Jennrich and M Schluchter Unbalanced repeated measures models with structured covariance matrices Biometrics 42 805 820 1986 I G G Kreft and J de Leeuw Introducing Multilevel Modelling Sage publications London Thousand Oaks New Delhi 1998 I G G Kreft J de Leeuw and K S Kim Comparing four different statis
17. el for two level data is 2a Yi By Log Bhuj ar E B pig E555 3We have changed the HLM notation to make it conform more to MLwiN notation 8 JAN DE LEEUW AND ITA G G KREFT where 2b B 78020 Ys12j1 F H YshZjh Usj To make the comparison with MLwiN somewhat easier we rewrite this as 3 Yi YooLoij2jo YphUpig2jh T Ups Loig T Ups Lpig Ej There are two obvious differences between the programs 1 In MLwiN we can have random coefficients Bog on the first level This is not possible in HLM 2 In HLM the emphasis is on the cross level interactions in the fixed part which are products of a first level and a second level predictor It is possible to have such variables in MLwiN but they are not as central 2 3 4 Interface Similar to what we saw in MLn the newer versions of HLM now have a Windows interface In addition they have the interac tive question and answer and batchfile interfaces from the older versions In Bryk et al 1996 the authors remark that most PC users will prefer the Windows interface but from the rest of the book it is pretty clear that they themselves do not The Windows version of HLM also comes with an equation editor similar to the one in MLwiN 2 3 5 Algorithm By default HLM 2L uses REML estimation while HLM 3L uses F IML Nevertheless both programs can actually do both forms of es timation Older versions of HLM relied on the EM algorithm which can some
18. evel Project is the book by Harvey Goldstein Not surprisingly the book also exists in two very different editions It developed with the project The first edi tion 1987 covers basic multilevel analysis with emphasis on applications in education The second edition 1995 is much more statistical covers a large variety of extensions of the basic model and a much broader range of applications 2 1 3 Model The basic model in the case of two levels and p predictors is Ja Yy Boi 015 T Bayti Pess b pijt Pii where It is easy to see how this generalizes to more than two levels Observe that usually we have xo 1 for all 7 7 i e the zero term in the regression corresponds with the intercept Also observe that each regression coeffi cient has a fixed part and a random part where the random part has random components for both levels 2 1 4 Interface As indicated above MLwiN provides or maybe we should say is a Windows interface on top of MLn It is possible to use MLwiN very much like MLn because there is a com mand window in which MLn commands can be entered This window also tracks the command history In fact menu commands are translated to MLn commands in the command window There are many MLn and NANOSTAT commands which cannot be entered from the menus An interesting feature of the program is the equations window This is a specialized equations editor in which the model can be defined in equation form
19. g in the background One does not need a GUI to prepare parameter files in fact this is not the natural way to prepare them at all If you are a command line program at heart then you should acknowledge that and you should not pretend that a layer of Windows make up can change this A command line interface which builds up the parameter file by asking questions which is the interface for VARCL and which was also the in terface for older versions of HLM has some advantages over the Windows interfaces It is obviously more structured and the user does not have to search around in the menus for the appropriate commands Strange and impossible combinations of options are more easily avoided It is more fool proof On the other hand it rapidly becomes very boring to go through the same sequence of questions every time and more experienced users will prefer to type the answers ahead of time into a command file and use the program in batch mode which makes the interface similar to the batch versions of MIXFOO or MLA If we summarize our findings there is little doubt that MLwiN is the most comprehensive program for multilevel analysis is Of course it is not ex actly fair to compare MLwiN with the whole SAS or even BMDP system Those are more comprehensive and they can also be used to perform mul tilevel analysis HLM obviously has also developed into a mature and ex cellent product Its basic design is more modest than MLwiN it lacks t
20. he macro language the graphics and the NANOSTAT statistics and data han dling MLA is somewhat limited but it can do many useful things some of which are impossible or very hard to do in the other packages Probably for most academic users the price of the packages is not re ally a problem The fact that MLA is free and that MLwiN costs money does not seem to be a major problem We must say however that the suit of MIXFOO programs by Hedeker is really a bargain One can do ordinary multilevel analysis multilevel survival analysis categorical and ordinal out comes Poisson multilevel analysis and all this in a uniform and modular way The programs are fast have little overhead are versatile well docu mented and completely free Moreover binaries are available for Power Macs and for Sun Solaris machines For most of the MIXFOO programs one needs to write batch setup files but as we have indicated above that is not really a major disadvantage Market forces have conspired to make us 20 JAN DE LEEUW AND ITA G G KREFT believe that it is and that graphical interfaces are the be all and end all of software but most quality statistical and numerical software does not have and does not need such interfaces Editing an existing text file is usually faster than wading through not very intuitive menus REFERENCES M A Aitkin and N T Longford Statistical modeling issues in school effectiveness studies Journal of
21. ince it includes all previous programs as special cases and since most people seem to feel that putting a Windows interface on top of a DOS program is a step ahead 2 1 1 Availability The Multilevel Project has three mirror homepages at www ioe ac uk multilevel www medent umontreal ca multilevel www edfac unimelb edu au multilevel The websites have information about the software but also about the project and its activities The MLwiN program has its own homepage at www ioe ac uk mlwin The price of a single copy given on the MLwiN website is 900 500 with a 40 educational discount 2 1 2 Documentation There is a large amount of documentation avail able most of it is can be obtained from the multilevel project Again the various versions of the program lead to a somewhat bewildering variety of documents Basically there are three types of documents available for most versions of the program The first document is the user s guide For MLwiN this is Goldstein et al 1998 There is another type of user s guide which is more introductory For Min this is Woodhouse 1995 The second is the advanced manual that discusses macros to fit more complicated mod els This is Yang et al 1998 Thirdly there is the command reference Since the command language for MLwiN is actually MLn this is Rasbash and Woodhouse 1995 SOFTWARE FOR MULTILEVEL ANALYSIS 5 In a sense the most important document in the Multil
22. l the minor program will become a major one It will become bigger have more possibilities look better and cost more 1 2 Omissions There are some programs that can be used to perform multilevel analysis but that we do not discuss in detail We just mention these programs briefly here and we give the URL in case readers want to know more TERRACE This is multilevel research software written by James Hilden Minton for his Ph D thesis It is an add on to the XLISP STAT pack age and it can be found with manual at www stat ucla edu consult paid nels papers NLME The NLME software comprises a set of S plus functions meth ods and classes for the analysis of both linear and nonlinear mixed effects models It extends the linear and nonlinear modeling facilities available in release 3 of S plus Written by Jos C Pinheiro and Douglas M Bates It is available for Unix and Windows platforms from franz stat wisc edu pub NLME BUGS BUGS is a piece of computer software which permits the analysis of complex statistical models using Markov Chain Monte Carlo meth ods The emphasis is on the Monte Carlo method and a great variety of multilevel models can be analyzed as well See www mrc bsu cam ac uk bugs mainpage html for further details Oswald Oswald developed by the Statistics Group at the University of Lancaster is a suite of S p1lus functions for analyzing longitudi nal data www maths lancs ac uk Software O
23. m the DOS command line as mla lt inputfile gt outputfile Not much of an interface but it does the job 3 3 5 Algorithm MLA uses the BFGS algorithm which is a general pur pose quasi Newton optimization algorithm to maximize either FIML or REML Alternatively it can also use EM In order to make sure that the level two dispersion matrix is positive semi definite two different parametriza tions are available that ensure this 14 JAN DE LEEUW AND ITA G G KREFT 3 3 6 Extensions MLA is somewhat unremarkable as a multilevel program although the parametrizations of the dispersion matrix and the use of BFGS are unique But it is remarkable because it supports a wide variety of simu lation analyses This tends to suggest it is research software but for practi tioners the confidence information will also be useful Here is a list of the unique features taken from the website e different kinds of simulations bootstrap jackknife and permutation e different methods of bootstrap simulation cases parametric and error e different types of residual estimation raw and shrunken e different cases resampling schemes level 1 level 2 and both e balanced resampling schemes balanced bootstrap e linking of residual levels e distribution plots histograms for parameters standard errors and t value 4 MODULES IN A MAJOR PACKAGE 4 1 BMDP5 V BMDP was the first major statistical package It has always had
24. oi Sp E B tai Eijs where Nis is the unobserved continuous response and 1 if ko lt n lt K 25 if ki lt N lt Ko Tb y A all Lij T if kr 1 lt n lt Kr where k oo and k 00 3 2 4 Interface All programs exist as batch versions using command files MIXOR and MIXREG also have interactive DOS version in which the com mand file is constructed from menu commands issued by the user Finally MIXREG and MIXOR are available as Windows programs 3 2 5 Algorithm MIXREG uses a combination of the EM and the scoring algorithm in much the same way as for instance HLM For MIXOR there are additional complications because multidimensional integrals must be evaluated to compute the likelihood and its derivatives MIXOR approxi mates these integrals by using Gauss Hermite quadrature 3 2 6 Extensions If we consider MIXREG and MIXOR to be the basic components then MIXGSUR MIXNO and MIXPREG are extensions But of course drawing the line in this way is rather arbitrary It is better to think of the whole set of programs as a modular alternative to programs such as SOFTWARE FOR MULTILEVEL ANALYSIS 13 HLM and MLwiN which cover about the same amount of territory in a single program 3 3 MLA MLA is a batch program running under DOS It is written by Frank Busing Rien van der Leeden and Eric Meijer of the University of Leiden The Netherlands It differs from
25. on matrices become singular or even indefinite 2 2 Extensions By extensions we mean various options and additions that do not really belong to the core of the program but that the authors have added because of user demand competitive pressure or their research program Some extensions are obviously more useful for the general public than others but most of them are at least interesting enough to be men tioned In MLwiN there are two levels of extensions The first are features which are part of the program core They can be handled by using the menus or the MLn language but they are features that will not often be needed The second level consists of true extensions written in the macro language provided by the package and these are add ons that one may or may not load The first class of extension are discussed in the user s guide Goldstein et al 1998 This already covers a substantial number of procedures Hi erarchical generalized linear models with binomial or Poisson outcomes can be fitted Monte Carlo Markov Chain methods are available to opti mize complicated likelihoods or compute complicated posterior distribu tions Parametric bootstrap methods are used for bias correction and for standard error computation These extensions may be exciting to some but they do take MLwiN sev eral steps in the direction of research software The user has to take very many things for granted and has to hope that the defaul
26. plicated territory she is invited to climb on the roller coaster close her eyes and enjoy the ride 2 3 HLM HLM s the history is similar to MLwiN s There first was a HLM for two level models then one for three level models then one which could also do generalized linear regression and finally a Windows version This last version version 4 of HLM 2L and HLM 3L is the one we review here Of course HLM 2L does two level analysis and HLM 3L three level anal ysis The programming was done by Richard Congdon the HLM team also includes Stephen Raudenbush and Tony Bryk 2 3 1 Availability The software is available from Scientific Software In ternational in Chicago where HLM has its homepage www ssicentral com hlm mainhlm htm The price is 400 for the DOS version and 430 for the Windows version 2 3 2 Documentation The HLM documentation consists of a user s man ual Bryk et al 1996 and of the book by Bryk and Raudenbush 1991 In a sense the book is independent of the software but since you get the book when you buy the software and since the HLM program is used through out the book the two are really intimately related even closer than MLwiN and Goldstein 1995 The choice of name HLM can lead to unfortunate confusion of the model the technique and the package de Leeuw and Kreft 1995 The Windows interface to HLM is only documented in the on line help it seems 2 3 3 Model The basic HLM mod
27. random coefficient model which allows for random slopes as well Thus random coefficients which seemed to be missing in the BMDP 5 V specification enter again through the back door Model 10d says that within subject covariance matrices have a factor analysis structure while 10e describes a general linear covariance structure and 10f allows for any 16 JAN DE LEEUW AND ITA G G KREFT matrix except that it must be the same over subjects The final two mod els allow for autoregressive structures similar to the ones we have seen in MIXREG BMDP5 V also allows you to define your very own structure provided you write a FORTRAN program to compute the structure and its derivatives Really enterprising users could fit a repeated measures model in which the within subject covariance structures satisfy a LISREL model for instance 4 1 4 Interface BMDP is a very modular package Each of the 44 pro grams can basically be used in a stand alone way This makes it quite dif ferent from SAS which is much more integrated Modularity produces a great deal of efficiency On the other hand BMDP is very very DOS It comes with BMDP DYNAMIC a somewhat optimistically named data and program managing module BMDP DYNAMIC has the familiar DOS type pseudo menus and pseudo windows It is fairly lightweight in terms of overhead 4 1 5 Algorithm There is a variety of algorithms available If you choose the FIML c
28. report by by Hedeker et al 1996 More generally there is a list of both theoretical and applied articles using MIXFOO at www uic edu hedeker works html 3 2 3 Model For MIXREG the model is 6a Yij QA0Woij a A QpWpij By Log SP eases B vais Eijs with 6b Bo ps Usj 12 JAN DE LEEUW AND ITA G G KREFT This is a straightforward two level random coefficient model very similar to what we have in VARCL The unique aspect is that we do not assume that 6c C eij Eki Og but we allow for a much more general parametric first level error covari ance structure More specifically the e can have an autoregressive AR 1 moving average MA 1 or autoregressive moving average ARMA 1 1 co variance structure a general stationary autocorrelation structure and even a special nonstationary one For MIXOR the model is the same except that we do not observe the y p we observe a multi category version generated by a threshold model Thus there are unknown cut off points assumed to be the same for all variables If Ye is below the first cut off we observe a 1 if it is between the first and second cut off we observe a 2 and so on Thus the model is basically the same as before but there are missing data because we only know that Yi is in a particular interval but we don t know where it is in that interval More precisely the model is 7a Nij QoWoij Farnet QpWpij By L
29. riterion you can use Fisher scoring Newton Raphson or gener alized EM In case of REML generalized EM and quasi scoring are available 4 1 6 Extensions Some of the options for the within subject covariance matrix are rather esoteric and could be considered extensions Certainly options for which you have to write the code yourself qualify as such 4 2 PROC MIXED SAS has many modules or PROC s to fit linear mod els and generalized linear models Until fairly recently however there was nothing that could compete with BMDP 5 V But now there is and with a vengeance PROC MIXED is the most general program of the ones we dis cuss it is also embedded in the totality of SAS or whatever parts of SAS you happen to have installed 4 2 1 Availability You can obviously order the package from SAS Institute Start at www sas com service techsup fag stat_proc mixeproc html Because SAS is such a huge and complicated product it does not make much sense to give a price Many users will simply add PROC MIXED to their PC or mainframe SAS setup Many people probably already have it available without actually knowing this SOFTWARE FOR MULTILEVEL ANALYSIS 17 4 2 2 Documentation The documentation for PROC MIXED is very ex tensive It consist of a chapter in SAS 1992 This is written in the familiar format of the SAS manuals It has almost 100 pages setup like a user s guide There is not much theory
30. similar program because it has extensive simulation possibilities built in notably the bootstrap and the jackknife and it has various ordinary least squares estimation methods as options 3 3 1 Availability Software and manual can be downloaded from www fsw leidenuniv nl www w3_ment medewerkers BUSING MLA HTM The program is free 3 3 2 Documentation The software distribution contains a 70 page Post script manual Busing et al 1994 It is a bit wordy because it tries to spell out all details especially the technical ones The ultimate example is the 20 page technical appendix A which gives in painstaking detail the derivations of the formulas for the likelihood function its first and second derivatives and their expectations It illustrates the effect TEX has on the mind of an individual who has just escaped from the dungeons of WYSIWYG The user s guide portion of the manual is quite clear however The authors have informed us that the manual is out of date because many options have been added to MLA in the meantime The program can now make histograms of bootstrap results for instance and scatterplots of residuals and there are different bootstrap based confidence intervals Also permutation tests for testing intra class correlation are available 3 3 3 Model The model is the same as in 2a and 2b 3 3 4 Interface MLA requires the user to create a parameter file and then the program is started fro
31. swald It includes mixed effects models and many other possible options 2 MAJOR SPECIALIZED PROGRAMS There are two major specialized programs for performing multilevel anal ysis one from the UK and one from the US Throughout we omit the http part of the URL s 4 JAN DE LEEUW AND ITA G G KREFT 2 1 MLwiN The ML series of programs has a complicated history The series was erected on top of the NANOSTAT program by Michael Healy NANOSTAT is a general purpose statistics program The multilevel exten sions started with ML2 in 1988 ML3 was introduced in 1990 and the final DOS program in the series was MLn published in 1995 In 1998 they were all superseded by MLwiN One can think of MLwiN as a separate program but also as a graphical user interface on top of MLn Throughout the project most of the programming was done by Jon Rasbash but clearly the program is the result of a team effort MLwiN contains the NANOSTAT package so it can do a fair amount of data manipulation and general purpose statistics This is all meant to assist in the multilevel analysis and thus we still think of MLwiN as a specialized program In our report Kreft et al 1990 we looked at ML2 in the published version Kreft et al 1994 at ML3 In a subsequent comparison the internal report version van der Leeden et al 1991 looked at ML3 the published version van der Leeden et al 1996 at MLn We shall only discuss MLwiN in this paper s
32. t op tionally one can start with some Fisher scoring iterations 4 2 6 Extensions The two major extensions of PROC MIXED are both written in the macro language SAS IML by Russ Wolfinger They are discussed in detail in Littell et al 1996 Ch 11 and 12 Generalized linear mixed models can be fitted with the GLIMMIX macro This incorporates binomial models with logit and probit links and Poisson count models with the log link The macros uses a joint quasi likelihood method Nonlinear mixed models can be fitted with the NLINMIX macro This macro actually has three options depending on whether first order or second order or marginal approximations are used This allows one to fit a really large range of models such as nonlinear growth curve models It is probably obvious that using such macros cannot possibly be very efficient Using SAS to fit your multilevel models is already quite a stretch because you have to carry along the enormous overhead of the SAS system A class of students each using SAS on a single CPU will show you what this means On top of that the use of macros and the SAS IML interpreter adds another layer of inefficiency It s a little bit like using a big truck to pick up some groceries at your local supermarket There is a trade off of course Many people have SAS installed and are quite familiar with its interface and macro language If they need to fit a multilevel model once in a while it probably does
33. t values of the many parameters and tuning constants work in her case It seems a bit too de manding to ask the casual user to choose between the MQL and PQL meth ods for quasi likelihood estimation or to choose an appropriate burn in pe riod for her MCMC procedure It is true that with these extensions models can be fitted that could not be fitted by older versions of MLn But for most people it becomes impossible to understand what is actually going on inside the program and to explain why certain choices and not others were made The macro based extensions of MLwiN allow for even more flexibility Macros are available for fitting multi category models survival models SOFTWARE FOR MULTILEVEL ANALYSIS 7 time series models and nonlinear models While these extensions are un doubtedly useful again the reservations in the previous paragraph apply For the average MLwiN user the instructions in the manual are voodoo There are references to the statistical literature of course but these refer ences are in many cases too technical to be of much use This of course is a well known dilemma If applied researchers with often quite limited technical expertise want to fit very complicated models with very complicated algorithms then the documentation and the imple mentation should handle this very carefully Both in Goldstein 1995 and in the various manuals one often gets the idea that instead of carefully guid ing the user through com
34. times be hopelessly slow It is now possible to speed up convergence by switching to Fisher scoring 2 3 6 Extensions There are a number of interesting extension in HLM First there is the V known option where the variance components are sup posed to be known which can be used in meta analysis Second there are hierarchical generalized linear models fitted by using penalized quasi likelihood or generalized estimation equations In particular Poisson Bernoulli and binomial models can be fitted Third a form of plausible value analysis using multiple imputation is available for two level models Although the scope of HLM is obviously much smaller than that of MLwiN this restriction of generality makes the program easier to use Generally the documenta tion is more user oriented the number of choices the user can make is more limited and often the authors of the program have already made many of the choices SOFTWARE FOR MULTILEVEL ANALYSIS 9 3 MINOR SPECIALIZED PROGRAM 3 1 VARCL The program VARCL started out in the mid eighties as one of the major contenders It was Longford s research software used in the path breaking paper by Aitkin and Longford 1986 and in his book Long ford 1993 Since 1990 Longford has moved to using S plus for research software development and nothing has happened with VARCL It is still a DOS program 3 1 1 Availability The program is sold by ProGAMMA in Groningen Nether lands but
35. tunate time interval between the two which is especially annoying in the case of software reviews The reviews were summarized briefly in our book Kreft and de Leeuw 1998 Section 1 6 Since our last publication on the subject there have been many major changes The program GENMOD which was never easy to obtain has more or less completely disappeared VARCL which was one of the leading con tenders in the early nineties is not actively supported or developed any more which means it rapidly lost ground BMDP as a company went out of business which had serious consequences for its software products Pro grams such as HLM written originally for DOS were upgraded for Win dows ML3 transformed to MLn which transformed to MLwiN And so on Obviously it is time for an update In this update we will change our focus somewhat We will indicate what the programs can do where you can get them for how much money on what systems they run and how easy it is to use them We do not emphasize computation speed because this is hard to define and not very relevant in most applications anyway Computing time is usually infinitely small compared to the time needed to collect and clean the data 1 1 Programs Macros and Packages Throughout this review we use a number of classifications of the software we review Of course the bound aries between the categories are somewhat fuzzy Research versus Production Some software is written
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